Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574100 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,10)(7,14)(8,11)(13,18)(15,16)(17,20);(1,2)(3,4)(5,6,11,18,9,13,8,10)(7,12,14,20,16,19,15,17);(2,4)(5,7)(6,12)(8,15)(9,16)(10,17)(11,14)(13,19)(18,20);(1,2)(3,4)(6,13)(10,18)(12,19)(17,20);(2,4)Polycyclic presentation:
2^9|b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
| maximal quotient of | #72322382, #72322424, #72322798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481538);